3.877 \(\int \frac{(A+B x) (a+b x+c x^2)^3}{x^8} \, dx\)

Optimal. Leaf size=160 \[ -\frac{a^2 (a B+3 A b)}{6 x^6}-\frac{a^3 A}{7 x^7}-\frac{3 a A c^2+6 a b B c+3 A b^2 c+b^3 B}{3 x^3}-\frac{3 a \left (A \left (a c+b^2\right )+a b B\right )}{5 x^5}-\frac{3 c \left (a B c+A b c+b^2 B\right )}{2 x^2}-\frac{A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{4 x^4}-\frac{c^2 (A c+3 b B)}{x}+B c^3 \log (x) \]

[Out]

-(a^3*A)/(7*x^7) - (a^2*(3*A*b + a*B))/(6*x^6) - (3*a*(a*b*B + A*(b^2 + a*c)))/(5*x^5) - (3*a*B*(b^2 + a*c) +
A*(b^3 + 6*a*b*c))/(4*x^4) - (b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)/(3*x^3) - (3*c*(b^2*B + A*b*c + a*B*c
))/(2*x^2) - (c^2*(3*b*B + A*c))/x + B*c^3*Log[x]

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Rubi [A]  time = 0.106111, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {765} \[ -\frac{a^2 (a B+3 A b)}{6 x^6}-\frac{a^3 A}{7 x^7}-\frac{3 a A c^2+6 a b B c+3 A b^2 c+b^3 B}{3 x^3}-\frac{3 a \left (A \left (a c+b^2\right )+a b B\right )}{5 x^5}-\frac{3 c \left (a B c+A b c+b^2 B\right )}{2 x^2}-\frac{A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{4 x^4}-\frac{c^2 (A c+3 b B)}{x}+B c^3 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^3)/x^8,x]

[Out]

-(a^3*A)/(7*x^7) - (a^2*(3*A*b + a*B))/(6*x^6) - (3*a*(a*b*B + A*(b^2 + a*c)))/(5*x^5) - (3*a*B*(b^2 + a*c) +
A*(b^3 + 6*a*b*c))/(4*x^4) - (b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)/(3*x^3) - (3*c*(b^2*B + A*b*c + a*B*c
))/(2*x^2) - (c^2*(3*b*B + A*c))/x + B*c^3*Log[x]

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (a+b x+c x^2\right )^3}{x^8} \, dx &=\int \left (\frac{a^3 A}{x^8}+\frac{a^2 (3 A b+a B)}{x^7}+\frac{3 a \left (a b B+A \left (b^2+a c\right )\right )}{x^6}+\frac{3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )}{x^5}+\frac{b^3 B+3 A b^2 c+6 a b B c+3 a A c^2}{x^4}+\frac{3 c \left (b^2 B+A b c+a B c\right )}{x^3}+\frac{c^2 (3 b B+A c)}{x^2}+\frac{B c^3}{x}\right ) \, dx\\ &=-\frac{a^3 A}{7 x^7}-\frac{a^2 (3 A b+a B)}{6 x^6}-\frac{3 a \left (a b B+A \left (b^2+a c\right )\right )}{5 x^5}-\frac{3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )}{4 x^4}-\frac{b^3 B+3 A b^2 c+6 a b B c+3 a A c^2}{3 x^3}-\frac{3 c \left (b^2 B+A b c+a B c\right )}{2 x^2}-\frac{c^2 (3 b B+A c)}{x}+B c^3 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0855866, size = 175, normalized size = 1.09 \[ -\frac{21 a^2 x (2 A (5 b+6 c x)+3 B x (4 b+5 c x))+10 a^3 (6 A+7 B x)+21 a x^2 \left (2 A \left (6 b^2+15 b c x+10 c^2 x^2\right )+5 B x \left (3 b^2+8 b c x+6 c^2 x^2\right )\right )+35 x^3 \left (3 A \left (4 b^2 c x+b^3+6 b c^2 x^2+4 c^3 x^3\right )+2 b B x \left (2 b^2+9 b c x+18 c^2 x^2\right )\right )-420 B c^3 x^7 \log (x)}{420 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/x^8,x]

[Out]

-(10*a^3*(6*A + 7*B*x) + 21*a^2*x*(3*B*x*(4*b + 5*c*x) + 2*A*(5*b + 6*c*x)) + 21*a*x^2*(5*B*x*(3*b^2 + 8*b*c*x
+ 6*c^2*x^2) + 2*A*(6*b^2 + 15*b*c*x + 10*c^2*x^2)) + 35*x^3*(2*b*B*x*(2*b^2 + 9*b*c*x + 18*c^2*x^2) + 3*A*(b
^3 + 4*b^2*c*x + 6*b*c^2*x^2 + 4*c^3*x^3)) - 420*B*c^3*x^7*Log[x])/(420*x^7)

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Maple [A]  time = 0.009, size = 192, normalized size = 1.2 \begin{align*} B{c}^{3}\ln \left ( x \right ) -{\frac{aA{c}^{2}}{{x}^{3}}}-{\frac{A{b}^{2}c}{{x}^{3}}}-2\,{\frac{abBc}{{x}^{3}}}-{\frac{{b}^{3}B}{3\,{x}^{3}}}-{\frac{3\,Ab{c}^{2}}{2\,{x}^{2}}}-{\frac{3\,a{c}^{2}B}{2\,{x}^{2}}}-{\frac{3\,{b}^{2}cB}{2\,{x}^{2}}}-{\frac{A{c}^{3}}{x}}-3\,{\frac{Bb{c}^{2}}{x}}-{\frac{A{a}^{3}}{7\,{x}^{7}}}-{\frac{3\,A{a}^{2}c}{5\,{x}^{5}}}-{\frac{3\,aA{b}^{2}}{5\,{x}^{5}}}-{\frac{3\,B{a}^{2}b}{5\,{x}^{5}}}-{\frac{3\,Aabc}{2\,{x}^{4}}}-{\frac{A{b}^{3}}{4\,{x}^{4}}}-{\frac{3\,B{a}^{2}c}{4\,{x}^{4}}}-{\frac{3\,Ba{b}^{2}}{4\,{x}^{4}}}-{\frac{Ab{a}^{2}}{2\,{x}^{6}}}-{\frac{B{a}^{3}}{6\,{x}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x+a)^3/x^8,x)

[Out]

B*c^3*ln(x)-1/x^3*a*A*c^2-1/x^3*A*b^2*c-2/x^3*a*b*B*c-1/3/x^3*b^3*B-3/2*c^2/x^2*A*b-3/2*c^2/x^2*a*B-3/2*c/x^2*
b^2*B-c^3/x*A-3*c^2/x*b*B-1/7*a^3*A/x^7-3/5*a^2/x^5*A*c-3/5*a/x^5*A*b^2-3/5*a^2/x^5*b*B-3/2/x^4*A*a*b*c-1/4*A*
b^3/x^4-3/4/x^4*B*a^2*c-3/4/x^4*B*a*b^2-1/2*a^2/x^6*A*b-1/6*a^3/x^6*B

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Maxima [A]  time = 1.07075, size = 223, normalized size = 1.39 \begin{align*} B c^{3} \log \left (x\right ) - \frac{420 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 630 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} + 140 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 60 \, A a^{3} + 105 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 252 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 70 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{420 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^3/x^8,x, algorithm="maxima")

[Out]

B*c^3*log(x) - 1/420*(420*(3*B*b*c^2 + A*c^3)*x^6 + 630*(B*b^2*c + (B*a + A*b)*c^2)*x^5 + 140*(B*b^3 + 3*A*a*c
^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 + 60*A*a^3 + 105*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 + 252*(B*a^2*
b + A*a*b^2 + A*a^2*c)*x^2 + 70*(B*a^3 + 3*A*a^2*b)*x)/x^7

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Fricas [A]  time = 1.40482, size = 389, normalized size = 2.43 \begin{align*} \frac{420 \, B c^{3} x^{7} \log \left (x\right ) - 420 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} - 630 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} - 140 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} - 60 \, A a^{3} - 105 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} - 252 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} - 70 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{420 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^3/x^8,x, algorithm="fricas")

[Out]

1/420*(420*B*c^3*x^7*log(x) - 420*(3*B*b*c^2 + A*c^3)*x^6 - 630*(B*b^2*c + (B*a + A*b)*c^2)*x^5 - 140*(B*b^3 +
3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 - 60*A*a^3 - 105*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 - 252
*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^2 - 70*(B*a^3 + 3*A*a^2*b)*x)/x^7

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Sympy [A]  time = 92.617, size = 184, normalized size = 1.15 \begin{align*} B c^{3} \log{\left (x \right )} - \frac{60 A a^{3} + x^{6} \left (420 A c^{3} + 1260 B b c^{2}\right ) + x^{5} \left (630 A b c^{2} + 630 B a c^{2} + 630 B b^{2} c\right ) + x^{4} \left (420 A a c^{2} + 420 A b^{2} c + 840 B a b c + 140 B b^{3}\right ) + x^{3} \left (630 A a b c + 105 A b^{3} + 315 B a^{2} c + 315 B a b^{2}\right ) + x^{2} \left (252 A a^{2} c + 252 A a b^{2} + 252 B a^{2} b\right ) + x \left (210 A a^{2} b + 70 B a^{3}\right )}{420 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x+a)**3/x**8,x)

[Out]

B*c**3*log(x) - (60*A*a**3 + x**6*(420*A*c**3 + 1260*B*b*c**2) + x**5*(630*A*b*c**2 + 630*B*a*c**2 + 630*B*b**
2*c) + x**4*(420*A*a*c**2 + 420*A*b**2*c + 840*B*a*b*c + 140*B*b**3) + x**3*(630*A*a*b*c + 105*A*b**3 + 315*B*
a**2*c + 315*B*a*b**2) + x**2*(252*A*a**2*c + 252*A*a*b**2 + 252*B*a**2*b) + x*(210*A*a**2*b + 70*B*a**3))/(42
0*x**7)

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Giac [A]  time = 1.35492, size = 223, normalized size = 1.39 \begin{align*} B c^{3} \log \left ({\left | x \right |}\right ) - \frac{420 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 630 \,{\left (B b^{2} c + B a c^{2} + A b c^{2}\right )} x^{5} + 140 \,{\left (B b^{3} + 6 \, B a b c + 3 \, A b^{2} c + 3 \, A a c^{2}\right )} x^{4} + 60 \, A a^{3} + 105 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \, B a^{2} c + 6 \, A a b c\right )} x^{3} + 252 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 70 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{420 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^3/x^8,x, algorithm="giac")

[Out]

B*c^3*log(abs(x)) - 1/420*(420*(3*B*b*c^2 + A*c^3)*x^6 + 630*(B*b^2*c + B*a*c^2 + A*b*c^2)*x^5 + 140*(B*b^3 +
6*B*a*b*c + 3*A*b^2*c + 3*A*a*c^2)*x^4 + 60*A*a^3 + 105*(3*B*a*b^2 + A*b^3 + 3*B*a^2*c + 6*A*a*b*c)*x^3 + 252*
(B*a^2*b + A*a*b^2 + A*a^2*c)*x^2 + 70*(B*a^3 + 3*A*a^2*b)*x)/x^7